In fact, \(\operatorname{Spec} \mathbb {F}\)  is the étale analogue of the Eilenberg-Maclane space \(K(\widehat{\mathbb {Z}}, 1)\)  (see [1]} for the definition of this notion). On the topological side, since \(\pi _1(S^1)\simeq \mathbb {Z}\)  and \(\pi _i(S^1)=0\)  for \(i\ge 2\) , the circle \(S^1\)  is \(K(\mathbb {Z}, 1)\) . For every integer \(n\ge 1\) , there is a unique extension \(\mathbb {F}^{\prime }/\mathbb {F}\)  of degree \(n\)  with \(\operatorname{Gal}(\mathbb {F}^{\prime }/\mathbb {F})\simeq \mathbb {Z}/n\mathbb {Z}\) . On the topological side, there is a unique \(\mathbb {Z}/n\mathbb {Z}\)  cover \([n]:S^1\rightarrow S^1\) . Identifying \(S^1\)  with the unit circle in \(\mathbb {C}\) , the covering map \([n]\)  maps \(z\mapsto z^n\) . Thus the finite Galois covers of \(\mathbb {F}\)  are in correspondence with finite covers of \(S^1\) .
